A Symmetrical Streamline Stabilization scheme for high advective transport

Abstract

SUMMARYAn overview of numerical techniques and previous investigations related to the solution of advectiondominated transport processes is presented. In addition a new Symmetrical Streamline Stabilization (S) scheme is introduced. The basis of the technique is to treat the transport equation in two steps. In the "rst step the dispersion part is approximated by a standard Galerkin approach, while in the second step the advection is approximated by a least-squares method. The two parts are reassembled, resulting in one system of equations. The resulting coe$cients' matrix is symmetric. Only half of a sparse matrix needs to be stored. Robust iterative algorithms for symmetrical systems of equations such as the preconditioned conjugate gradient method (PCG) can be successfully used. The new method leads to an implicit introduction of an &arti"cial di!usion' term. Solute transport with high Peclet and Courant numbers does not lead to oscillations due to an inherent upwind damping. The upwind e!ect acts only in #ow direction. The e$ciency of the new formulation in terms of accuracy and computation time is shown in comparison with the Galerkin approach for mesh parallel and mesh oblique high advective solute transport.

Modelling of Advection-Dominated Transport in a Porous Column with a Time-Decaying Flow Field

“…It therefore has the common factor α v · ∇w (where α is called upwind function or intrinsic time of the stabilized methods (Oñate et al, 1997); v is the characteristic velocity and w is a weighting function). The perturbation parameter, α, can be chosen based on either the Least Squares Methods, such that the artificial convection term has the adjoint form of a convection term in the equation and consequently the numerical scheme becomes symmetric (Wendland and Schmid, 2000), or be based on a Fourier analysis to ensure that numerical modelling can give an "optimal" solution for the transient advection-dispersion equation (Raymond and Garder, 1976), such as the Streamline Upwind PetrovGalerkin Method (Hughes and Brooks, 1982). It can also take different values for the temporal term from the spatial term in the equation to generate different stabilized methods, such as the Taylor-Galerkin method.…”

“…The expression for the upwind function, α i (i = 1, 2), for several stabilized finite element methods, the Petrov-Galerkin method (PG) (Hughes and Brooks, 1982), the second and third-order Taylor-Galerkin method (TG2 and TG3) (Donea et al, 1984), the Least Squares method (LS) (Carey and Jiang, 1987) as well as the Modified Least Squares Method (MLS) (Wendland and Schmid, 2000;Selvadurai and Dong, 2005) are listed in the Table III, where h is the element length, t is the time step and the associated Courant Number is Cr = |v| t h. It can be seen from the residual integral Equation (19) that the perturbation in the asymmetric weighting function will introduce not only an "artificial diffusion" term but also an "artificial convection" term in the discretized form of the equation.…”

“…Certain key contributions in this area include the works of Raymond and Garder (1976), Hughes and Brooks (1982), Vichnevetsky and Bowles (1982), Donea et al (1984), Morton (1996) and Oñate et al (1997). Examples that evaluate the computational performance of schemes used to model both advective and advective-dispersive flows in single and double porosity media are given by Zhao and Valliappan (1994), Valliappan et al (1998), Wendland and Schmid (2000) and Wang and Hutter (2001).…”

On nonclassical analytical solutions for advective transport problems

“…The Symmetric Streamline Stabilization technique proposed by Wendland and Schmid [2000] for strongly advection dominated processes, introduces an upwind term into the least squares technique to obtain a better computational performance in advection‐dominated processes. This is equivalent to using different perturbation parameters in the least squares scheme for the temporal and spatial terms of the advection equation.…”

A Taylor–Galerkin approach for modelling a spherically symmetric advective–dispersive transport problem

“…The fact that the ETG scheme gives accurate solutions for the advection equation and unstable solutions for the advection-dispersion equation indicates that an operator-splitting procedure needs to be coupled with the ETG scheme to develop an accurate solution for the advection-dispersion equation [23]. Figure 6 illustrates the oscillation-free and non-diffusive numerical solutions, obtained from the operator-splitting ETG scheme, for the one-dimensional advective-dispersive transport process considered in the previous section.…”